Finding Roots Of Quadratic Function F(x) = (x - 4)(x + 2)
In the realm of mathematics, quadratic functions hold a significant position, serving as the foundation for numerous applications across various fields. These functions, characterized by their parabolic curves, are defined by the general form f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are constants. One of the fundamental aspects of understanding quadratic functions lies in identifying their roots, also known as zeros or x-intercepts. These roots represent the points where the parabola intersects the x-axis, providing crucial insights into the function's behavior and properties. In this comprehensive guide, we will delve into the intricacies of finding the roots of a specific quadratic function, f(x) = (x - 4)(x + 2), and explore the underlying concepts that govern this process.
Understanding the Significance of Roots
Before we embark on the journey of finding the roots of our given function, let's take a moment to appreciate the significance of these values. The roots of a quadratic function provide valuable information about its graph, the parabola. Specifically, they indicate the points where the parabola crosses the x-axis. These points are crucial for sketching the graph, determining the function's minimum or maximum value, and solving related equations and inequalities. Moreover, roots play a pivotal role in real-world applications, such as modeling projectile motion, optimizing engineering designs, and analyzing financial trends.
Factoring the Quadratic Function
Our given quadratic function, f(x) = (x - 4)(x + 2), is presented in a factored form. This form provides a direct pathway to finding the roots. The factored form of a quadratic function expresses it as a product of two linear expressions, each representing a factor of the quadratic. In our case, the factors are (x - 4) and (x + 2). The key principle here is the Zero Product Property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property forms the cornerstone of our root-finding process.
To find the roots, we set the function equal to zero: f(x) = (x - 4)(x + 2) = 0. Applying the Zero Product Property, we set each factor equal to zero: x - 4 = 0 and x + 2 = 0. Solving these simple linear equations, we obtain x = 4 and x = -2. These values represent the roots of the quadratic function. Geometrically, these roots correspond to the points where the parabola intersects the x-axis, namely (4, 0) and (-2, 0).
The Roots and the Parabola's Intercepts
The roots of a quadratic function hold a direct connection to the parabola's intercepts with the x-axis. The x-intercepts are the points where the parabola crosses the x-axis, and these points have a y-coordinate of zero. Consequently, the x-coordinates of the x-intercepts are precisely the roots of the quadratic function. In our example, the roots are 4 and -2, which means the parabola intersects the x-axis at the points (4, 0) and (-2, 0). These points provide valuable reference points for sketching the graph of the parabola.
Visualizing the Roots on the Graph
To solidify our understanding, let's visualize the roots on the graph of the quadratic function. The graph of f(x) = (x - 4)(x + 2) is a parabola that opens upwards, as the coefficient of the x^2 term is positive. The roots, 4 and -2, mark the points where the parabola intersects the x-axis. These intersections divide the x-axis into three regions. In the region to the left of -2, the function's values are positive. In the region between -2 and 4, the function's values are negative. And in the region to the right of 4, the function's values are again positive. This sign behavior is a characteristic feature of quadratic functions and is directly linked to the location of the roots.
Alternative Methods for Finding Roots
While factoring provides a straightforward approach for finding the roots of our specific function, other methods exist for solving quadratic equations. These methods become particularly useful when factoring is not readily apparent or when dealing with more complex quadratic expressions. Two prominent methods are the quadratic formula and completing the square.
The quadratic formula is a universal tool for finding the roots of any quadratic equation. It provides a direct solution regardless of whether the equation can be factored easily. The formula is expressed as: x = (-b ± √(b^2 - 4ac)) / 2a, where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. Applying the quadratic formula to our function, we would first expand the factored form to obtain x^2 - 2x - 8 = 0. Then, we identify a = 1, b = -2, and c = -8. Substituting these values into the quadratic formula, we would arrive at the same roots, x = 4 and x = -2.
Completing the square is another powerful technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial on one side, which can then be easily factored. While completing the square can be more involved than factoring or using the quadratic formula, it provides valuable insights into the structure of quadratic equations and can be advantageous in certain situations.
The Discriminant and the Nature of Roots
An important concept related to the roots of a quadratic function is the discriminant. The discriminant is the expression b^2 - 4ac, which appears under the square root in the quadratic formula. The value of the discriminant reveals the nature of the roots – whether they are real or complex, distinct or repeated.
If the discriminant is positive, the quadratic function has two distinct real roots. This corresponds to the parabola intersecting the x-axis at two distinct points. If the discriminant is zero, the quadratic function has one repeated real root. This corresponds to the parabola touching the x-axis at a single point, the vertex of the parabola. If the discriminant is negative, the quadratic function has two complex roots. This corresponds to the parabola not intersecting the x-axis at all.
In our example, the discriminant is (-2)^2 - 4 * 1 * (-8) = 36, which is positive. This confirms that our function has two distinct real roots, as we found earlier.
Real-World Applications of Roots
The roots of quadratic functions find widespread applications in various real-world scenarios. In physics, they are used to determine the trajectory of projectiles, such as the path of a ball thrown into the air. The roots represent the points where the projectile hits the ground. In engineering, quadratic functions are employed to optimize designs, such as the shape of a bridge arch or the dimensions of a container. The roots can help identify the optimal values that maximize strength or minimize material usage. In finance, quadratic functions can model investment growth and predict the break-even points for businesses. The roots represent the points where the investment reaches a certain target value or where the business becomes profitable.
Conclusion
Finding the roots of quadratic functions is a fundamental skill in mathematics with far-reaching implications. By understanding the relationship between roots, factors, and the graph of the parabola, we gain a deeper appreciation for the behavior of these functions. In this guide, we explored the process of finding the roots of f(x) = (x - 4)(x + 2) using factoring and discussed alternative methods, such as the quadratic formula and completing the square. We also examined the significance of the discriminant in determining the nature of the roots and highlighted real-world applications where roots play a crucial role. With this comprehensive understanding, you are well-equipped to tackle a wide range of problems involving quadratic functions and their roots.
The heart of the question lies in finding the values of x that make the function f(x) = (x - 4)(x + 2) equal to zero. These values are known as the roots or zeros of the function, and they represent the points where the graph of the function intersects the x-axis. Understanding how to find these roots is crucial for analyzing the behavior of the function and its applications in various fields.
Understanding the Zero Product Property
The key to solving this equation lies in the Zero Product Property. This fundamental principle states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In our case, the function f(x) is expressed as the product of two factors: (x - 4) and (x + 2). Therefore, to find the roots, we need to determine the values of x that make either of these factors equal to zero.
Setting Each Factor to Zero
Following the Zero Product Property, we set each factor equal to zero and solve for x:
- x - 4 = 0
Adding 4 to both sides of the equation, we get:
x = 4
- x + 2 = 0
Subtracting 2 from both sides of the equation, we get:
x = -2
The Roots of the Function
Therefore, the roots of the function f(x) = (x - 4)(x + 2) are x = 4 and x = -2. These values represent the x-coordinates of the points where the graph of the function intersects the x-axis.
Expressing the Roots as Coordinates
The roots can also be expressed as coordinate pairs. Since the roots are the points where the function equals zero, the y-coordinate of these points is 0. Therefore, the roots can be represented as the following coordinates:
- (4, 0)
- (-2, 0)
Matching the Roots to the Answer Choices
Now, let's compare our findings to the answer choices provided:
- A. (0, 2) - Incorrect
- B. (-2, 0) - Correct
- C. (-4, 0) - Incorrect
- D. (4, -2) - Incorrect
The Correct Answer
As we can see, the correct answer is B. (-2, 0), as this coordinate pair represents one of the roots we calculated.
Expanding the Function
To further illustrate the concept, let's expand the function f(x) = (x - 4)(x + 2):
f(x) = x^2 + 2x - 4x - 8
f(x) = x^2 - 2x - 8
This is the standard quadratic form of the function. We can use this form to verify our roots by plugging them back into the function and confirming that the result is zero.
Verifying the Roots
Let's verify the roots x = 4 and x = -2:
- For x = 4:
f(4) = (4)^2 - 2(4) - 8
f(4) = 16 - 8 - 8
f(4) = 0
- For x = -2:
f(-2) = (-2)^2 - 2(-2) - 8
f(-2) = 4 + 4 - 8
f(-2) = 0
As we can see, plugging both roots into the function results in zero, confirming that they are indeed the roots of the function.
Graphical Representation
The graph of the function f(x) = x^2 - 2x - 8 is a parabola that intersects the x-axis at the points (-2, 0) and (4, 0), which are the roots we calculated. This visual representation further solidifies our understanding of the roots and their relationship to the function.
The Significance of Roots
In summary, the roots of a function are the values of x that make the function equal to zero. They represent the x-intercepts of the graph of the function and play a crucial role in understanding its behavior and applications. The Zero Product Property is a powerful tool for finding the roots of functions expressed in factored form.
In conclusion, by applying the Zero Product Property and setting each factor of the function f(x) = (x - 4)(x + 2) to zero, we successfully identified the roots as x = 4 and x = -2. These roots correspond to the points (4, 0) and (-2, 0) on the graph of the function. Comparing these results to the answer choices provided, we determined that the correct answer is B. (-2, 0). This exercise demonstrates the fundamental importance of understanding the Zero Product Property and its application in finding the roots of functions.
